In a large pot, cover the potatoes with cold water. Season with salt. Bring to a boil and let simmer for 20 minutes or until tender. Drain and return to the pot. With a potato masher, crush the potatoes with the butter. Gradually add the milk and purée until smooth. Season with salt and pepper. Keep warm.
With the rack in the middle position, preheat the oven to 400°F (200°C).
In a large skillet over medium heat, soften the green onions in the butter. Add the cream and spinach. Cook until the spinach has wilted and reduce the cream until almost dry. Season with salt and pepper. Transfer to an 11 x 8-inch (28 x 20 cm) baking dish.
Spread the mashed potatoes over the spinach. With a spatula or the back of a spoon, smooth the potatoes into a wave pattern. Top with the cheese slices. At this stage, the dish may be refrigerated.
Bake for 20 minutes (or 40 minutes if refrigerated). Finish cooking under the broiler for 2 to 3 minutes or until golden brown. Let rest for 10 minutes.
In a bowl, combine the melted butter and herbs. Spoon the parsley butter over the gratin.